Use another example to simplify.
He’s not so much acting unintelligently as acting uninterested in schoolwork.
While the subject is actively unintelligently, what is more important is his lack of interest.
So for your three examples:
1. He acts not so much unintelligently as uninterested.
2. He does not act so much unintelligently as uninterested.
3. He does not act unintelligently so much as uninterested.
3 sounds the most natural, but the meaning is clear in all 3. In each case, both the first and the second descriptions are true, but the emphasis is on the second.
And just to add a bit more complexity, the following statements are also possible:
It's not just that he drives quickly, as much as drives recklessly.
It's not that he drives quickly, as much as that he drives recklessly.
It's not his driving aggressively that's the issue, it's his recklessness while driving.
It's not his aggressive driving, it's his recklessness while driving that's the issue.
As always depends on what exactly you are trying to express. Figure out your primary intention with this statement, and that should help guide your word choice.
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